Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-7x+6y &= 1 \\ -x+4y &= -1\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-x = -4y-1$ Divide both sides by $-1$ to isolate $x$ $x = {4y + 1}$ Substitute this expression for $x$ in the first equation. $-7({4y + 1}) + 6y = 1$ $-28y - 7 + 6y = 1$ Simplify by combining terms, then solve for $y$ $-22y - 7 = 1$ $-22y = 8$ $y = -\dfrac{4}{11}$ Substitute $-\dfrac{4}{11}$ for $y$ in the top equation. $-7x+6( -\dfrac{4}{11}) = 1$ $-7x-\dfrac{24}{11} = 1$ $-7x = \dfrac{35}{11}$ $x = -\dfrac{5}{11}$ The solution is $\enspace x = -\dfrac{5}{11}, \enspace y = -\dfrac{4}{11}$.